Electronically Aiding Students\u27 Learning

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Interactive Qualifying Projects (All Years) Interactive Qualifying Projects

May 2008

Electronically Aiding Students' Learning

Adam Lee Schwartz

Worcester Polytechnic Institute Andrew V. Tsitsilianos Worcester Polytechnic Institute Sarah Elizabeth Judd Worcester Polytechnic Institute

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Electronically Aiding Students’ Learning

An Interactive Qualifying Project Report:

Submitted to the Faculty of the

WORCESTER POLYTECHNIC INSTITUTE In partial fulfillment of the requirements for the

Degree of Bachelor of Science By

__________________________ __________________________ Sarah Judd Andrew Tsitsilianos

__________________________ Adam Schwartz

Date: May 9, 2008 Advisors:

__________________________ Professor Neil Heffernan

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Abstract……….………..3 Introduction………4 Previous System ...……….5 Hints ………...8 Scaffolding Questions ………...…..10 Worked Examples………....22 Knowledge components ………..…25 Appendix A: Hints Only

Appendix B: Scaffolding Questions Appendix C: Worked Examples Appendix D: Wiki Pages

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Abstract:

The Assistments System was designed by Professor Neil Heffernan to help students learn. We have been enhancing the Assistments system to explore the benefits of scaffolding questions, worked examples, and explanations of individual knowledge components as a means to the main goal. Hopefully, the work we have done these past few terms will translate to more informed middle school students in the subject area of math.

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Introduction:

The Massachusetts Comprehensive Assessment System, or MCAS, must be passed by all public high school students in 10th grade in order to graduate. The questions are written by a board of teachers to best represent of the curriculum frameworks designed by the Massachusetts Department of Education. This recently added requirement test students in mathematics, English language arts, science and technology, history, and social sciences. The tests are an array of multiple-choice, short-answer, and open-response questions along with essay sections in the English and

language arts portion. Because of the growing concern of students not passing, Professor Neil Heffernan of Worcester Polytechnic Institute has developed a web-based tutoring program for mathematics called Assistments.

Assistments tested two different types of learning methods, hints and scaffolding. A scaffolding problem breaks a problem into smaller, more basic questions or

sub-problems that break down the original problem. Each of those basic questions has its own hints guiding them in the correct direction. Hints simply gave hints a sequential order if the student requested them, eventually leading them to the final answer. The sub-problems were designed to be easy to understand for a student, using bright colors to highlight and emphasize areas of importance. Assisstments enables each student to essentially receive their own tutor, because the system walks the student through each problem. It also allows the student to work at their own pace without the pressure of

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circulation around Worcester. A key component which is currently of great assistance to the teachers is the knowledge component.

A knowledge component is a way to tag a problem based on the type of math required to solve the problem, for example, geometry, addition, etc. These were written by our group and can accessed through a wikipage. Eventually, there will be a link from the assistments page to the wikipage to assist the student in that area of concern with a general definition, how it’s used, and an example. There are currently 106 knowledge components. Assigning each problem the proper knowledge component is extremely valuable for the teacher, which in turn benefits the student. At any time, the knowledge components aide teachers by allowing them to easily asses what areas need work for individual students or as a whole class. This will help teachers manage their class time and use it more efficiently, directly assessing students needs.

The purpose of this project was to first write hints or scaffolding problems from previous MCAS tests or investigation activities for the 8th grade. After which, all the knowledge components were written by our team on the wikipage.

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Currently, Assissments is in its second generation. The first generation was quite successful, but did have many bugs and problems, which were constantly being updated. The questions provided by the assistment system were recreated from previous MCAS tests. For the first system, knowledge components were not implemented to the extent of they are now. As questions were designed, the builder tagged what type of knowledge component applied, but most went untagged until after the first system was out of use. Two different types of questions were designed, scaffolding questions and hint questions. The original assistment system was not user friendly compared to the new system. The save time when writing problems took at times up to five minutes, which turned into a large amount wasted time. The MCAS question was written in the main question with an option to upload a picture. The answer section required a response type, for example multiple choice.

One downside to the 1st generation system was that each time any part of the problem was changed, the problem needed to be saved, which equated in too much time spent on save time. The save time problem was later fixed with an autosave feature which was added to the next system.

The first system was designed to determine the effectiveness of the two different type of learning strategies, hints or scaffolding. The data from the answered questions was complied into an excel file and a method was developed to determine whether learning had occurred. This turned out to be a very tough task because it was tough to determine if the students had actually learned from the help provided. Although

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the time and most questions went without being tagged to the detail that was required. Although the use of knowledge components was not prominent in the first system, the usefulness was discovered and in the later system applied much more vigorously. Currently, the knowledge components are separated into 106 different categories based on the type of math required to solve the problem. They are organized by the different categories with the percentage of students who answered each question correctly with the question also available for viewing. The teacher can also browse by class period or student. Having all problems tagged with knowledge components to the new

assistment system allows the teacher to easily identify which area the students need help in individually or as a class. The knowledge components also aides the teacher because it allows them to understand what areas they might have trouble teaching in which can allow them to improve their curriculum for the following year.

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Within the Assistment, there is a feature called hints. Once enabled, the Assistment writer is allowed to give little clues on what they question is asking. They unfold one after another until the last hint, which we usually put as the answer to the question asked.

They are pretty simple to create. A feature in the Assistment builder lets you select whether you want hints in the problem on which you are working. Once the hints are enacted, it is as simple as writing in the box, and saving. In the early system, hints were rudimentary; they gave you no special features like text fonts or colors. One would have to use html coding to implement any fancy writing.

With the new Assistment system, this was changed, to better facilitate the writers of the Assistments. All the text features one could ask for were given on a toolbar above the hint writing space. These features ranged from text font changes, to text color, and, of course, Bold, Italic, and Underline buttons. A table feature was also added, and with it, the ability to line up text, for keeping an equation looking neat or keeping values in place. The biggest addition, however, was the image uploader. It was unthinkable up until now to put an image in a hint; with the new feature, this dream became reality. Hints became much more stylish. We could now indicate things we would want the student to see in color or Bold. We could link those colors to pictures, graphs, tables, anything our imaginations could fathom. The addition of pictures, colors to the text, and all these special features did nothing but add to the feature of the hint.

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to give help. As the student would do the problem, if they got the question wrong, or didn’t want to give an answer, for fear of getting it wrong, they would click “Request Help”. This would then bring the hint up in a yellow box, with a suggestion on a way of completing the problem. Hints, however, didn’t see their real glory until scaffolding questions were enacted by the Assistment system. As before, you would click on the “Request Help” button, and look at the hints to the problem. However, in the scaffolding questions, hints became a sort of special way of breaking down the problem, simply and elegantly. With each successive scaffold or worked example, the hints within them provided a step by step track, a roadmap, to solving the question. This system is a great way to convey how one can do the problem given, and though it is still utilized, its essence paved the way for the ultimate hint system: The Assistment Knowledge Components.

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What they are:

Scaffolding breaks down a problem into its component questions. If the

problem involves knowing the Order of Operations, a Scaffolding question would first ask the student to come up with what that order would be. Form there, it

would ask the student knowing that, which part of the problem they should answer

first.

Scaffolding questions enable an electronic version of the Socratic method of

teaching. In person, the Socratic Method forces a teacher to ask questions rather than giving answers. In The Socratic Method: Teaching by Asking instead of by

Telling Rick Garlikov describes “the Socratic method in what [he] consider[s] its

purest form,” to be “where questions (and only questions) are used to arouse

curiosity and at the same time serve as a logical, incremental, step-wise guide that

enables students to figure out about a complex topic or issue with their own

thinking and insights.” (Garlikov) Scaffolding questions do just that, asking

logical, incremental questions that enable students to figure out the more complex original problem.

Students are walked through the problem, answering easier questions. This

allows them to see that they've understood the information since the beginning,

and teaches them what they didn't already know. Scaffolding questions force

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Socratic education.

How they relate to Hints:

Oftentimes, the students just can’t come up with the proper answers in a

timely enough fashion to move the lesson along. This frustrates the students

instead of exciting their curiosity and interest. At this point, a live teacher would

use verbal hints to give their students more direction. Eventually, the answer

would have to come from the teacher if it cannot come from the students. In a

similar manner, Assistments does not force a student into endlessly answering a scaffolding question that seems impossible to the student. Assistments also

provides the students with hints for each individual scaffolding question; the last

of which spells the answer out for the student so that the student can move on to

the next question. This mixes all the benefits of hints with the benefits of

scaffolding questions. Implementing scaffolding questions compounds the

benefits of hints.

Scaffolding questions do have definite advantages over hints alone, however, as proven by a study done by Leena Razzaq using the Assistments

program. She discovered that students’ performance improved using scaffolding

questions a significant amount more than using hints alone. This corresponded to

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slower learning process rather than a “get this done as quick as I can” process

benefits the students.

Scaffolding questions force more investment than hints on the part of the

student. A student can click hints, barely read them and find the answer to a

question. Scaffolding questions, on the other hand, require the student to interact

with what they’re learning while they’re learning it. The famous Chinese proverb

applies here: “I hear and I forget. I see and I remember. I do and I understand." At their bare minimum, hints only cover the “I hear” part. They tell the students

something that will hopefully lead to them getting the answer. Oftentimes, we try

to put pictures in our hints, illustrating how something works. This covers the “I

see and I remember.” Scaffolding questions, on the other hand, have the students

“do” something. They answer questions for themselves. Scaffolding questions

help the students reach the “I do and I understand” part of the proverb,

implementing centuries-old wisdom with technology.

Scaffolding Questions Benefit to teachers:

With computing comes the ability to keep track of more data. Assistments

takes full advantage of database technology to keep track of several minute details

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problems individual students got wrong as well as which broader learning areas the greatest percentage of students in a class had trouble with. It also displays how

many hints it took in each scaffolding question for a student to discover the

answer. This data can be essential for teachers who want to figure out which

topics they need to cover in more depth.

Because scaffolding questions are pieces of the original question, looking at

how many hints it took students to figure out individual scaffolding questions

pinpoints exactly where in the problem students are having trouble. On a problem about Order of operations, for example, the first scaffolding question usually asks

“What is the order of operations.” If students seem to get this question right, but

then struggle when implementing it, the teacher can focus on working out

examples on the board, and reviewing arithmetic. If the students are having

trouble with this first scaffolding question, however, the teacher knows that he or

she should go back to covering exactly what the order is. This allows more

effective use of class time. The teacher does not need to spend class time going over each individual piece of a problem in depth, he or she can spend depth time

only on the precise location the student needs it. Scaffolding questions, combined

with Assistments power to collect, store, and sort information can be a powerful

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Oftentimes, a student understands the components of the problem, but is

confused when seeing everything at once. Breaking the question down builds confidence, as the students see the pieces of the question that they understand.

Scaffolding questions are also an effective review mechanism. If a student

has fallen behind in a particular area, that student can use the scaffolding questions

as an in depth review of the topic. The VARK Learning style model created by

Fleming explores the fact that different students have different learning styles.

The Kinesthetic learning style would benefit from Scaffolding questions

especially. Fleming’s description of this learning style’s intake strategy is “The ideas on this page are only valuable if they sound practical, real, and relevant to

you.” (Fleming) These students get the least benefit from a traditional read and

lecture style, and so need the most attention they can get inside their learning style.

It would also seem that a computer program built around asking questions and

getting answers from reading material would similarly turn off these students, but

scaffolding questions assure that this is not the case. Scaffolding questions put the

material its teaching into a relevant context. The student has been asked to answer a question that he or she was unable to understand. The scaffolding questions

relate directly back to that question. Kinesthetic learners who need to connect

utility with what they’re learning will pay more attention, and thus learn more.

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and their own answers, thus also benefiting from scaffolding questions. While aural learners do not benefit from the silent Assistments program, they are the

most likely to have understood directly from the lectures themselves, and thus will

probably bypass these scaffolding questions to begin with. Scaffolding questions

appeal to almost all learning styles, making them an effective review tool.

Our implementation:

Developing scaffolding questions is an art. As a general rule, we decided

each problem should be associated with three scaffolding questions. This allows the student to move through the problems at a reasonable pace, while still looking

into depth at each problem.

We did not always follow this rule, however. Sometimes we would come

across a question that would bring up an important point in the scaffolding

questions themselves. One example of this was a problem where the student using

the program was given a fraction of problems correct for each of the problem’s

students. Each of these fractions should have been easily converted into recognizable percents for the student (3/4 by sixth grade should be trivial to

convert to 75%). If they had gotten the problem wrong, it might have been

because the fraction to percent conversion had not been recognizable. To

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conversions weren’t quick, it was worth the student’s time to learn them. Only

after these questions was the comparison mentioned. The goal of these

scaffolding questions became not only to help the student answer the original

problem, but to fix a gap in their knowledge, making it worth the students time to

answer more than 3 scaffolding questions.

On the other hand, we would occasionally run into a problem where while

the idea behind the original problem was important, less than three scaffolding questions would more than cover the topic. One example includes a problem

asking how a made-up student could check her answer to a subtraction problem.

The answer the MCAS had been looking for was to check your answer against the

reverse operation, using fact families. This is a useful thing for a student to know;

even the brightest students make mistakes, and knowing the tricks to catch them

could change a student’s grade or MCAS score. To slowly egg the answer out of a

student in three scaffolding questions, however, would drag the student’s time out unnecessarily. We would prefer them spend time on the question described in the

previous paragraph. Because of this, we gave the students only one scaffolding

question to answer- a similar question using easier numbers where the relations

between numbers became more obvious. Three scaffolding questions per problem made for a nice rule, but only when it didn’t interfere with assuring the students’

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out broad and move until we actually give the students the answer. This circumvents the problem of students getting stuck on one portion, while not

immediately giving them an answer when all they want is a hint. See the

discussion under scaffolding questions: connection to hints.

The very last scaffolding question in every group restates the initial

problem. The hints on this scaffolding question often referred back to the

connection between the scaffolding questions they had already been answering to

this question, further cementing the ideas into the student’s head. This rounds out the scaffolding questions, reminding the students why they had learned the

previous material. The scaffolding questions should have hit upon the trouble the

student had with the initial problem. Reiterating that problem gives the student a

second chance at it, understanding more of the information. Discovering that they

now understand the question builds the students’ confidence in their newly found

knowledge.

To encourage this confidence, we try not to answer the entire question for the student in the scaffolding questions. A question on order of operations, for

example, might ask what the order of operations are, and follow it up with a

question asking about the first step of the initial problem. We would not,

however, continue in that vein until the problem is completely solved. This allows students to continue applying their knowledge to actually solving the problems for

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Hints for development:

In order for the scaffolding questions to work, they need to keep the

students attention. The explanations in the hints must be short and understandable

enough that even students whose first language isn’t English can follow it. Using

colors for certain ideas aid in the ease of following explanations. Pictures

illustrate an idea, rather than writing it out in a complicated manner. Keeping

individual lines short keeps the students’ eyes moving across the page, soaking in the details of individual lines rather than the entire page. We make sure the style

of each individual assistment eases the students learning ability, rather than

frustrating them.

How the Builder Works:

A teacher who wants to build an assistment logs on to Assistments, and

clicks build. They then click “Create new Assistment”. From there, they write the

original problem and save it. If they want hints and only hints, they enable hints.

If they want scaffolding, they enable scaffolding. Only one or the other will work

for each individual question. They can then click the type of problem: fill in, algebra (where ! is the same as 0.75), or multiple choice. After this, they enter in

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however, buggy messages are not necessary, as getting the question wrong will automatically enter the scaffolding questions if scaffolding is enabled. Each of

these scaffolding questions work exactly like the original question. One can also

click a trash can to delete the section they are looking at.

Suggestions for improvement:

assistment developers:

For now, the people writing “assistments” for the program attend WPI and

are receiving IQP credit for doing so. We have the professor who started the

program and the Reasearch Assistants and MQP students working on the software

available to us to explain things, fix bugs, and aid in writing of individual

problems. As the program gets less and less buggy, however, we hope to allow

teachers to build their own assistments. Students are getting credit for time spent

working on assistments, teachers have responsibilities to run class and grade papers and don’t have the time that students do. More teachers will be convinced

to use the Assistments program the easier and faster developing problems can be.

Developing for the program for the past few months have led to insights to what

might make this easier for teachers.

Several questions a teacher might want to add to Assistments would be similar to each other with different numbers. These questions are called “morphs”

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in one sweep of copy and paste would speed up this process immensely. A form

that gives the teacher the option to completely copy an entire assistment with

certain words or numbers replaced with words and numbers specified in a

particular form would also make the teachers life easier.

When first getting used to the Assistments program, deleting more than

was meant to be deleted can be easy to do, for example entire problem when all

that needed to be deleted was one particular scaffolding question. People think they know what they are deleting when they hit delete. A dialog box asking “Are

you sure you want to permanently delete this item” would not save people from

entirely deleting a half an hours worth of work they want to keep. Having that

message say exactly what you are deleting – “Are you sure you want to delete

scaffolding question one or scaffolding question beginning with the first line of

that scaffolding question” vs. “Are you sure you want to delete this assistment”

would have be more useful messages.

This problem would not have been so vital if retrieval of deleted items was

at all possible. The space this would take up on the server would be worth it.

Space could be conserved if Assistments servers only saved the last delete for

every user. This deletes the purposeful deletes from the server-if the teacher has moved on to a new question, he/she probably is no longer interested in than

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Students using Assistments

Picture a sunny day, the first time the rain has stopped all week, and a

swing set begging to be played upon. Students do not, as a general rule, want to

be answering that beg, not working on math problems. The longer students have to be working on something, the less attention they will want to pay to it.

Scaffolding questions take time to complete. If used to gain understanding, this is

still time well spent. If the student has accidentally typed the wrong answer,

however, this is not time well spent. If the student realizes their mistake on the

first scaffolding question, the other two questions are not time well spent. Continually finding themselves caught answering questions that they actually

understood, and knew that they understood will frustrate students, putting them

into the mindset of “just get this done as quickly as possible” which we already

know detriments their learning rather than the mindset of understanding. If

students could return to the original question at any point during the scaffolding

question process, it would eliminate time wasting questions so that the student

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What they are:

In a worked example, the student first sees a similar question to the one

they are currently working on. They then get to see how this problem can be solved. The idea is, they will be able to understand why this method worked and

how it did what it did, then be able to translate this to the problem they are

working on directly.

Worked examples are based on the “cognitive load theory” suggested by J

Sweller. Short term memory can only hold so much information at once. Long term memory puts together “schemas” that make one element out of several,

connecting ideas, and thus can hold more. Cognitive overload occurs when too

many individual things attempt to enter the short term memory at once. J. Sweller

suggests that learning occurs by building schemas as early as possible, so that

many concepts can be taken into the short term memory as one concept.

I used this method extensively in person when teaching middle school

students robotics. Programming a their LEGO Mindstorm robots to follow a black line required a different way of thinking. As an “expert” programmer, relative to

these students at least, I already had several schemas around if statements, while

statements, and the usage of sensors by the time I learned the trick. As “novice”

programmers, they were still developing these basic schemas. To aid them in the

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Teaching them the whole thing, then having them teach it back to me, turned the multiple concepts they needed to know in order to implement the line following

program into one basic schema, speeding up the learning process for these

students.

Worked examples in Assistments bring this idea to a virtual environment.

They keep all the information a student needs in one spot, explaining concepts in

one problem that allows the students to focus their energies on the learning aspects

– structures and applications of rules – rather then on individual questions.

How they relate to Scaffolding questions:

Scaffolding questions focused on making the problem easier for the student

by breaking it down. Worked examples make the problem easier by creating a

schema for it, connecting the individual broken down pieces. Scaffolding

questions built the student’s confidence by showing them that they understood the

material. Worked examples build confidence by trusting the student to make

his/her own connections. Currently, other IQP teams are looking to see whether scaffolding questions or worked examples work better for the students. The

results will be used to better aid students in learning.

The information displayed in multiple scaffolding questions are compiled

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Benefit to Students:

Worked examples engage students in learning by forcing them to take an

active approach to making connections. Rather than forcing the student to answer

several questions in pieces, worked examples give students the opportunity to

understand the applications of the rules. Learning these applications brings the

information into long term memory quicker, speeding up the learning process.

Our Implementation:

Worked examples work very similarly to scaffolding questions as far as the Assistments program is concerned. If a student gets a problem wrong, or requests

help, we bring up a worked example. The student must click a button saying that

they have read and understood the problem before they go on to retry the problem.

When creating these worked examples, we again paid close attention to

formatting so that the student was not overwhelmed with black, wordy

information. We made sure to continue our use of colors and pictures and carriage

returns, as this was even more important when all the information was presented at once than it previously had been.

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As with any method of teaching, sometimes even the best worked examples

were not all that a student needed. They might not have seen the connection between the worked example and their own problem, to unable to build a schema

for themselves. scaffolding questions, we gave the students hints on top of the

questions, even for the very last question, if they needed to be pushed along a little

further. Our model for worked examples at the current moment, however, does

not allow for that little extra push. I think we should include hints for the final

question of a worked example the same way we include them for our scaffolding

questions. This would cement the ideas we hoped to implement into all students’ heads, including those for whom the connection between the rules in the worked

example and the rules in the actual question was not as obvious.

Knowledge Components:

The knowledge component system is the final realization of teaching students within the Assistment pages. Using the wiki format, which was first designed by

Google, a website document writing style comparable to a cross between Microsoft word and html, we created tutorials of each component, which would be accessible through a link on relevant Assistment pages. These components use data collected from classes using the old and new Assistment system, which were compartmentalized into

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format, we decided to use a simple, clear cut explanation of each topic. Included in this explanation is a definition or process to each component, as well as examples of

problems the students would most likely see on an MCAS exam. The following figure is an example of what a finished page would look like from the writers prospective:

The wiki page gave us free reign on what types of interactive material we would want to use. The design we implemented was succinct yet informative, ideal for a student who would need to quickly brush up on the skill they would be using in the

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creating their own Knowledge Components for whatever suits their class needs.

Currently the wiki page can be accessed by going to the Assistment website then clicking Teacher Support Pages/Teacher Manual link. Once here, scroll down to the bottom and click 8th grade skills. Here is a list of all the knowledge components available. Professor Heffernan’s team is currently working on installing links in the problems for the desired knowledge component pertaining to each problem. An example of a page from the students perspective can be seen in the following figure:

This will be a great way for the students to get a basic knowledge of the material with simple examples, and a definition before they try to approach the problem at hand. Having these links directly in the problems should help the students learn by gaining

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Works Cited

Cooper, Graham. "Cognitive Load Theory as an Aid for Instructional Design." Ascilite. 1990. 5 May 2008

<http://www.ascilite.org.au/ajet/ajet6/cooper.html>.

Garlikov, Rick. "Richard Garlikov." The Socratic Method:. 27 Apr. 2008 <http://www.garlikov.com/Soc_Meth.html>.

Razzaq, Leena, and Neil Heffernan. Scaffolding Vs. Hints in the Assistment System. Diss. Worcester Polytechnic Institute.

Soloman, Howard. "Cognitive Load Theory." Explorations in Learning & Instruction:. 5 May 2008 <http://tip.psychology.org/sweller.html>.

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15774 15775 15844 15792 Teacher wiki: Area Combinatorics Equilateral triangle Fractions Graph Shape Integers

Least Common Multiple Measurement Use Ruler Number Line

Of Means Multiply Ordering Numbers Pythagorean Theorem

Rate with Distance and Time Reduce Fraction

Similar Triangles

Sum of Interior Angles Triangle Supplementary Angles Symbolization Articulation Andrew: Hints only: 12375 Scaffold: 12327 25874 12347 12330 12376 12395 12358 Teacher Wiki Stem and Leaf Plot Mean Median Order of Operations Circle Graph Sarah: Scaffold: 25703

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25727 25729 Worked Examples: 26255 26151 26153 26161 26150 Teacher wiki: Adding Decimals Addition Divisibility

Equivalent Fractions Decimals Percents Exponents

Fraction Division Fraction Multiplication

Making Sense of Expressions and Equations Multiplication

Multiplying Decimals

Multiplying Positive Negative Numbers Ordering Decimals Range Rounding Square Root Subtracting Decimals Subtraction Surface Area Venn Diagram

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Assistment #15774 (Pretty Print)

About People Press Funding Papers Contact

Home [FAQ]

Logout Mr Schwartz You are currently at: Assistment Pretty-Print

Edit this Assistment | Preview this Assistment

Hints: OnOn Buggies: OnOn Answers: AllAll KCs: OnOn Refresh "#2, Comparing and Scaling: Investigation 3 (2007/01/15 17:12:56)" (Problem ID: 15774) RADIO_BUTTON

No knowledge components have been assigned

Ben's dog can eat 5 bags of dog food in 24 days and Sandy's dog can eat 8 bags of dog food in 42 days. Jill's dog eats 3 bags of dog food in 12 days. Bennetts dog eats 11 bags of dog food in 50 days which dog eats the most dog food on average?

Answers: (Interface Type: RADIO_BUTTON) Ben's Dog

Bennett's Dog

Jill's Dog

Sandy's Dog Hint 1:

First figure out the average amount each dog eats in 1 day because it is not easy to compare how much each dog eats right now.

Hint 2:

To do this, divide the amount of bags the dog eats by the number of days it takes to eat those bags. Bags/Days = Amount of bags per day.

Hint 3:

Ben's Dog - 5/24 = 0.21 bags/day Sandy's Dog - 8/42 = 0.19 bags/day Jill's Dog - 3/12 = 0.25 bags/day Bennet's Dog - 11/50 = 0.22 bags/day Hint 4:

Now determine which dog has eaten the most each day. Hint 5:

Since Jill's dog has eaten .25 bags/day on average and this is larger that the amount eaten by the other dogs on average the answer is Jill's dog. Please select this answer.

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Assistment #15775 (Pretty Print)

[FAQ] Mr Schwartz

You are currently at: Assistment Pretty-Print

Hints: OnOn Buggies: OnOn Answers: AllAll KCs: OnOn Refresh

"#3, Comparing and Scaling: Investigation 3 (2007/01/15 17:37:07)" (Problem ID: 15775) TEXT_FIELD No knowledge components have been assigned

A store has a sale on sox. 3 Pairs of sox for $6.90 (you can buy one pair of sox for the sale price). If you have $10, how many pairs of sox can you buy?

Answers: (Interface Type: TEXT_FIELD) 4

Hint 1:

First you should figure out how much it costs for one pair of sox. Hint 2:

To do this, divide the cost for 3 pairs of sox by the number of pairs for the price. $6.90/3 = $2.30 for each pair of sox.

Hint 3:

Now figure out how many pairs of sox you can buy with $10 at $2.30 per pair. Hint 4:

To do this, divide the amount of money you have by the cost for each pair of sox. $10/$2.30 dollars = 4.35 pairs

Hint 5:

Remember that pairs of sox can only be bought in whole amounts, you can't buy .35 pair of sox. Hint 6:

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Assistment #15844 (Pretty Print)

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"#5, Scaling and Comparing: Investigation 4 (2007/01/17 20:48:33)" (Problem ID: 15844) ALGEBRA_FIELD No knowledge components have been assigned

Apples are on sale. You can buy 5 apples for $2. How much will it cost for 36 apples? Answers: (Interface Type: ALGEBRA_FIELD)

14.4 Hint 1:

Set up a proportion to help you solve this problem. Hint 2:

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Hint 3:

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Assistment #15792 (Pretty Print)

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Hints: OnOn Buggies: OnOn Answers: AllAll KCs: OnOn Refresh "#3, Comparing and Scaling: Investigation 4 (2007/01/16 18:44:36)" (Problem ID: 15792) RADIO_BUTTON

No knowledge components have been assigned

Which proportion can be used to calculate x? Answers: (Interface Type: RADIO_BUTTON)

A B C D

Hint 1:

The red sides correspond and the two different colored green sides correspond. Hint 2:

Now, set up a proportion to relate the similar triangles. Hint 3:

Remember the whole leg is 19. Hint 4:

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Hint 5:

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Assistment

You are previewing content.

There are 11 teachers and 132 students at a middle school. What is the ratio of teachers to students?

Comment on this question

A ratio is a comparison of two numbers. We are comparing the number of teachers to the number of students.

Comment on this hint

The ratio of teachers to students is 11 to 132, but this is not the simplest form.

Since 11 and 132 share a common factor, you should reduce the ratio to its simplest form.

11 and 132 share a common factor: 11. Divide both numbers by 11 to get the simplest form.

11/11 = 1 132/11 = 12

The simplest form of the ratio is 1 to 12. Choose B.

Comment on this hint Select one: A. 1 to 11 B. 1 to 12 C. 11 to 12 D. 11 to 13 Submit Answer Correct!

You are done with this problem!

Comment on this problem

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Assistment

Which of the following bestrepresents the location of point A on the numberline shown below?

Request Help Select one: A B C D Submit Answer

Let's move on and figure out this problem

First, lets get comfortable with the number line.

What is the value at the red dot?

The dot on a number line is placed over the number of its value.

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B is placed right over -1. The answer is -1. Type -1

Type your answer below:

Submit Answer Correct!

Now lets look at place on the number line that is not a whole number

What is the value of the red dot and point C?

The area in blue is between 1 and 2

Note that the line is broken into fourths.

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The correct answer is

C

Comment on this hint Select one: A B C D Submit Answer Correct!

Which of the following bestrepresents the location of point A on the numberline shown below?

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The following picture shows where each of the fractions on the number line are.

The answer is D

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Assistment

A sea otter has over 1,000,000 hairs per square inch on its back. Which of the following equals 1,000,000?

Comment on this question Request Help Select one: A. 105 B. 106 C. 107 D. 108 Submit Answer

Exponents are a way to show how many times a number is multiplied by itself. The example above shows 2 with an exponent of 3.

Which of the following equals 100? Comment on this question

You are looking for an expression that means 10 * 10 Comment on this hint

102 means 10 * 10 Comment on this hint Choose C. Comment on this hint

Select one: A. 100 B. 101 C. 102 D. 1010 Submit Answer Correct!

Good. Notice that 10 with an exponent of 2 gives you an expression with 2 zeroes at the end: 100.

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Now let's try the original question again.

Which of the following equals 1,000,000? Comment on this question

Notice the pattern: 101_{= 10}

102_{= 100} 103_{= 1000} 104_{= 10,000} Comment on this hint

1,000,000 has 6 zeroes so the answer should have an exponent of 6 Comment on this hint

Choose B Comment on this hint

You are done with this problem! Comment on this problem

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Assistment

MS. Patterson divided the students in her class into groups of 6 for a classroom activity. There were 2 students left over. Which of the following could be the number of students in Ms. Patterson's class? Comment on this question

Request Help Select one: 11 20 36 45 Submit Answer

Lets try to make sense of this situation.Lets say Ms Patterson had one group of six, with two extra students.

How many students would be in the class with one group of six and two extra people?

There is one group of six students.

1*6 = 6

Now add the two extra students

The total number of students in the class with one group of six and two extras is:

1 * 6 + 2 6 + 2 Comment on this hint

The total number of students in the class with one group of six and two extras is:

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6 + 2 8

The answer is 8. Type 8. Comment on this hint

Remembering how we solved the last question:

1*6+ 2= 8

Lets try it again for two groups of six.

How many students would be in the class with two groups of six and two extra people? Comment on this question

There are two groups of six students. 2 * 6 = 12

Now add the other two students Comment on this hint

The total number of students in two groups of six students with two extras can be solved by: 2 * 6+ 2

12 + 2

The total number of students in two groups of six students with two extras can be solved by:

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2 * 6+ 2 12 + 2

14

The answer is 14. Type 14.

Remembering how we solved the last question:

# of groupswork # of students in class 1 * 6 + 2 = 8 2 * 6 + 2 = 14

Use what we did above to solve the intial problem.

Ms. Patterson divided the students in her class into groups of 6 for a classroom activity. There were 2 students left over. Which of the following could be the number of students in Ms. Patterson's class? Comment on this question

# of group work # of students in cla 1 * 6 + 2 = 8 2 * 6 + 2 = 14 3 * 6 + 2 =

Lets continue the table we've been creating. The first number represents the number of groups of six students.

This is what the table would look like for three groups of six students with two extra students

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Comment on this hint # of groupswork # of students in the class 1 * 6 + 2 = 8 2 * 6 + 2 = 14 3 * 6 + 2 = 20

Three groups of six students with two extra students

is 20 students Comment on this hint

20 students is also a possible choice for the number of students in Ms. Patterson's class. The correct answer is twenty. Select B. 20

Comment on this hint Select one: A. 11 B. 20 C. 36 D. 45 Submit Answer Correct!

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Assistment

Carolyn and Kim are selling Lemonade this summer. It costs $0.10 to make each cup of lemonade.

They are going to sell each cup of lemonade for $0.25

If they sell 55 cups of lemonade, how much more money would they collect than they would spend? Comment on this question

Request Help

Type your answer below (mathematical expression):

Submit Answer

To figure out how much more money Carolyn and Kim would make than they would spend, lets first figure out how much money they would recieve from selling 55 cups of lemonade.

If Carolyn and Kim sell 55 cups of lemonade, what is the amount of money they will collect? Comment on this question

If Carolyn and Kim sell 1 cup of lemonade, at $0.25 they collect $0.25

1* .25 = .25 Comment on this hint

If Carolyn and Kim sell 2 cups of lemonade, at $0.25 each they collect $0.50

1 * .25 = .25

2 * .25 = .50 Comment on this hint

If Carolyn and Kim sell 55 cups of lemonade at $0.25, they collect $13.75

1 * .25 = .25

2 * .25 = .50

55 *.25 = 13.75

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Of the $13.75 Carolyn and Kim collect, some of it went into making the lemonade.

How much does it cost in dollars to make 55 cups of lemonade? Comment on this question

It costs $0.10 to make 1 cup of lemonade at $0.10 each. 1 * .1 = .1

It costs $0.20 to make 2 cup of lemonade at $0.10 each. 1 * .10 = .10

2 * .10 = .20 Comment on this hint

It costs $5.50 to make 55 cup of lemonade at $0.10 each. 1 * .10 = 0.10

2 * .10 = 0.20 55 * .10 = 5.50

The answer is $5.50 type 5.50 Comment on this hint

Now that we have seen two main parts finished, lets look at the original problem

13.75

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Carolyn and Kim are selling Lemonade this summer. It costs $0.10 to make each cup of lemonade.

They are going to sell each cup of lemonade for $0.25

If they sell 55 cups of lemonade, how much more money would they collect than they would spend?

In a problem like this you subtract the cost of making the lemonade from the earnings you recieve

If Carolyn and Kim had only made 1 cup of lemonade at a cost of $0.10 and sold it at a cost of $0.25 they would have earned $0.15

1 *.25 - 1* .25 =.15

If Carolyn and Kim had only made 2 cups of lemonade at a cost of $0.10 and sold it at a cost of $0.25 they would have earned $0.30

1 *.25 - 1* .10 =.15

2 *.25 - 2* .10 =.30

If Carolyn and Kim made 55 cups of lemonade at a cost of $0.10 and sold it at a cost of $0.25 they would have earned $8.25

1 *.25 -1 * .10 =0.15

2 *.25 -2 * .10 =0.30

55 * .25 -55 * .10 =8.25

The answer is $8.25. Type in 8.25 Comment on this hint

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Assistment

Each jar contains and equal number of coins.

The total number of coins in 6 of the 8 jars is 54. How many coins are in all 8 jars? Comment on this question

Request Help

Submit Answer

To find how many coins there are total, we first must find how many coins there should be in the two remaining jars. Let’s start by finding how many coins are in one jar.

How many coins should there be in the one jar? Comment on this question

The problem tells us that all the jars have the same number of coins in each one. Comment on this hint

6 jars contain 54 coins. That means that in 6 jars, there are 54 coins. Comment on this hint

To find how many coins there should be in each jar, you must divide the number of coins by the number of jars.

54 / 6 = 9 coins in one jar. Type in 9 Comment on this hint

Type your answer below (mathematical expression): 9

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Correct!

Now that we found how many coins are in each jar, we can find how many coins are in the two remaining jars.

How many coins are in the two remaining jars? Comment on this question

We have 2 jars, so you have to multiply the number of coins in one jar number by 2. Comment on this hint

9 * 2 = 18 Type in 18 Comment on this hint

Now lets return to the original problem.

How many coins are in all 8 jars? Comment on this question

There are 54 coins in 6 jars.

We found that there are 18 coins in the two remaining jars. Comment on this hint

Now we add the number of coins in 2 jars to the number of coins in 6 jars. Comment on this hint

Enter 72 18

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Assistment

On Kelly's homework, the answer to the following problem shown below was marked wrong 178-59 = 129

Which of the following is one way for her to discover that her answer is wrong? Comment on this question

Request Help Select one: 129 - 59 = 70 129 + 59 = 188 178 + 129 = 307 178 + 59 = 237 Submit Answer

Let's move on and figure out this problem Lets look at an easier version of this problem:

On John's homework, he got the following question wrong: 5 - 3 = 4

Which of the following is one way to show his answer is wrong? Comment on this question

Consider fact families, which show the relationships between numbers. In this case:

5 - 3 = 2 5 - 2 = 3 2 + 3 = 5 3 + 2 = 5 Comment on this hint

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Notice how if you add the answer to a subtraction problem to the number you're subtracting you get the intial number

5 - 3 = 2 2 + 3 = 5

If this does not happen, something is wrong Comment on this hint

John's problem reads 5 - 3 = 4

but

4 + 3 does not equal 5, it equals 7 Comment on this hint

The correct answer is that John can tell he is incorrect by realizing 4 + 3 = 7, not 5. The answer is B Comment on this hint

Select one: A. 4 - 3 = 1 B. 4 + 3 = 7 C. 4 + 5 = 9 D. 5 + 3 = 8 Submit Answer Correct!

On Kelly's homework the answer to the following problem shown below was marked wrong 178-59 = 129

Which of the following is one way for her to discover that her answer is wrong? Comment on this question

Consider fact families, which show the relationship between numbers. In this case:

178 - 59 = 119 178 - 119 = 59 59 + 119 = 178 119 + 59 = 178

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Notice how if you add the answer to a subtraction problem to the number you're subtracting you get the intial number

178 - 59 = 119 119 + 59 = 178

If this does not happen, something is wrong Comment on this hint

Kelly's problem reads 178 - 59 = 129

but

129 + 59 does not equal 178, it equals 188 Comment on this hint

the correct answer is that Kelly can tell that she is wrong by realizing that 129 + 59 = 188, not 178. the correct answer is B

Select one: A. 129 - 59 = 70 B. 129 + 59 = 188 C. 178 + 129 = 307 D. 178 + 59 = 307 Submit Answer Correct!

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Assistment

The grid below is shaded to represent a fraction.

What fraction of the grid is shaded?

Request Help Select one: A B C D Submit Answer

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the square?

Look at the following picture:

note how each red rectangle has the same number of small squres as the shaded one

There are 5 of these rectangles. The correct answer is 5

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Now lets go back to the original question.

The grid below is shaded to represent a fraction.

What fraction of the grid is shaded?

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A fraction is a part out of a whole. We have

The correct answer is

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Assistment

Judith has a total of 8 fish in her aquarium. Exactly 6 of the fish are guppies.

What percent of the fish in the aquarium are guppies?

Request Help Select one: A. 48% B. 60% C. 68% D. 75% Submit Answer

Let's first find the fraction of guppies in the fish tank compared to the total number of fish in the fish tank. Don't forget to reduce the fraction.

What fraction of the fish are guppies?

There are 8 fish in all. 6 out of 8 are guppies.

6/8 of the fish are guppies. You can reduce 6/8 further.

6/8 = 3/4. Type in 3/4

Now use the fraction to solve the orignial problem.

3/4

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Judith has a total of 8 fish in her aquarium. Exactly 6 of the fish are guppies.

What percent of the fish in the aquarium are guppies?

1/4 is 25%. What is 3/4 as a percent?

3/4 = 1/4 + 1/4 + 1/4 = 25% + 25% + 25% = ?

25% + 25% + 25% = 75% Choose D.

Comment on this hint Select one: A. 48% B. 60% C. 68% D. 75% Submit Answer Correct!

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Assistment

Which of the following is closest to the product 298.7 * 10.1?

Request Help Select one: A. 300 B. 2,000 C. 3,000 D. 20,000 Submit Answer

We are looking for an estimate, so that the numbers are easy to multiply. What is the best estimate of 298.7?

Choices A: 200 and C: 250 are not the best numbers with which to estimate because they are not the closest estimations to 298.7.

Choice B: 290 is still not the best estimation to 298.7 because it is not the closest number of the choices and it is also not as "nice" a number to work with.

Choice D: 300 is the best estimation because it is the closest number to 298.7 of the choices. Choose D.

Good. Now select the best estimate for 10.1?

Choices D: 1 can't be used because it is not close to 10.1.

Choice C: 11 is farther from 10.1 than 10.5, so it is not a good estimate.

Choice B: 10.5 is both farther from 10.1 and is not as "nice" a number as 10.

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Choice A: 10 is the best estimate for 10.1 because it is the closest number to 10.1 of the choices. Choose A.

Select one: A. 10 B. 10.5 C. 11 D. 1 Submit Answer Correct!

Now let's look at the original problem.

Which of the following is closest to the product 298.7 * 10.1?

We estimated 298.7 to 300 and 10.1 to 10.

Now it is easy to estimate 298.7 * 10.1 by multiplying 300 * 10 instead.

To multiply 300 * 10 just add a zero to 300 making it 3000. Therefore C is the best estimate. Select C.

Select one: A. 300 B. 2000 C. 3000 D. 30,000 Submit Answer Correct!

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Assistment

Shing made the design shown above using gray square tiles and white square tiles.

What fractional part of the whole design is made up of gray tiles? Write your answer as a fraction.

Request Help

Submit Answer

A fraction is part of a whole. Let's count how many tiles make up the whole design.

How many square tiles make up the whole design?

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First, count all of the tiles, white and grey.

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There are 25 tiles in all. Type in 25

Now let's count the grey tiles.

How many gray tiles are in the design? 25

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Count the grey tiles.

There are 15 gray tiles. Type in 15

Now let's try the original question.

What fractional part of the whole design is made up of gray tiles? Write your answer as a fraction. 15

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Write your fraction like this: gray tiles/all tiles.

There are 15 gray tiles and 25 tiles in all.

Type in 15/25

You can also reduce it to 3/5

Comment on this problem

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Assistment

Which of the following shows the numbers in order from least to greatest? Comment on this question

Request Help Select one: A. 0.765, 0.82, 0.791 B. 0.765, 0.791, 0.82 C. 0.791, 0.82, 0.765 D. 0.791, 0.765, 0.82 Submit Answer

Decimal numbers are another way to write fractions or mixed numbers. For example, the number above is four hundred seventy-nine and fifteen thousandths.

Let's look at another example: 35.907

What digit is in the tenths place in 35.907? Comment on this question

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The digit in the tenths place is 9. Choose C Comment on this hint

Select one: A. 3 B. 5 C. 9 D. 0 E. 7 Submit Answer Correct!

Let's look at the list of numbers.

Which number has the greatest value? Comment on this question

Line up the decimal points for the three numbers. Compare the digits in the greatest place first. Comment on this hint

The digits in the ones place are all the same, so compare the digits in the tenths place Comment on this hint

8 tenths is greater than 7 tenths, so 0.82 is the greatest number. Comment on this hint

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Comment on this hint Select one: A. 0.82 B. 0.765 C. 0.791 Submit Answer Correct!

Okay. 0.82 has the greatest value. Now we are left with 2 numbers: 0.765 and 0.791.

Which number is greater? Comment on this question

The ones digits and the tenths digits are the same. Compare the hundredths digits.

The hundredths digits are 6 and 9. Which one is greater?

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Comment on this hint Select one: 0.765 0.791 Submit Answer Correct!

Good. Now let's answer the original problem.

Which of the following shows the numbers in order from least to greatest? Comment on this question

0.82 is the greatest number. It should be last. Comment on this hint

0.791 is the next greatest number. It should be in the middle. Comment on this hint

0.765 is the smallest number. It should be first Comment on this hint

B. 0.765, 0.791, 0.82 shows the numbers in order from least to greatest. Choose B. Comment on this hint

Select one: A. 0.765, 0.82, 0.791 B. 0.765, 0.791, 0.82 C. 0.791, 0.82, 0.765 D. 0.791, 0.765, 0.82 Submit Answer Correct!

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Assistment

What is the value of the following expression?

3 + 6 * 4 Comment on this question

Request Help

Submit Answer

According to the correct order of operations, one operation should be done before the other.

Which operation must be done first? Comment on this question

There is a certain order that you must follow to find the value of a mathamatical expression. Comment on this hint

Just follow PE(MD)(AS):

Parenthesis, Exponents, Mulitplication and Division (from left to right), Addition and Subtraction (from left to right)

Take a look at the expression:

3 + 6 * 4

You see that there is multiplication and addition to do. Comment on this hint

You must do the multiplication first. Select Multiplication. Comment on this hint

Select one:

Multiplication

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Great, the multiplication should be done first.

Now let's look at the original problem.

3 + 6 * 4

What is the value of the expression? Comment on this question

Do the multiplication first.

3 + 6 * 4 Comment on this hint

Next, add the answer you got from the multiplication to the number left in the expression.

3 + 6 *4 3 + 24 Comment on this hint

3 + 6 *4 3 + 24 27

The value of the expression is 27. Type in 27 Comment on this hint

You are done with this problem! 27

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Assistment

Steve scored 1,086,000 points in a video game. Which of the following expressions below is equal to 1,086,000?

Comment on this question Request Help Select one: 100 + 80 + 6 1,000 + 80 + 6 1,000,000 + 80,000 + 6,000 1,000,000 + 800,000 + 60,000 Submit Answer

Let's move on and figure out this problem What does the 1 in 1,086,000 represent?

The 1 in 1,086,000 represents the same thing as the 1 in 1,234,546 and 1,567,300 Comment on this hint

The 1 in 1,086,000 also represents the same thing as the 1 in 1,000,000 Comment on this hint

The correct answer is 1,000,000. Comment on this hint

Select one: 100 1,000 1,000,000 Submit Answer Correct!

What does the 8 in 1,086,000 represent?

The 8 in 1,086,000 represents the same thing as the 8 in 85,235 and 183,000

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The 8 in 1,086,000 represents the same thing as the 8 in 80,000 Comment on this hint

The correct answer is 80,000 Comment on this hint

Select one: 80 8 80,000 8,000,000 Submit Answer Correct!

What does the 6 in 1,086,000 represent?

The 6 in 1,086,000 means the same thing as the 6 in 6,540 and 126,300 Comment on this hint

The 6 in 1,086,000 represents the same thing as the 6 in 6,000 Comment on this hint

The answer is 6,000 Comment on this hint Select one: 6 60 6,000 60,000 6,000,000 Submit Answer Correct!

Going back to the original problem:

Steve scored 1,086,000 points in a video game. Which of the following expressions below is equal to 1,086,000?

We know that the 1 in 1,086,000 stands for 1,000,000 We also know that the 8 stands for 80,000 and the 6 stands for 6,000. Consider what happens when you add these numbers

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1,000,000 + 80,000 + 6,000 = 1,086,000 The answer is C.

Comment on this hint Select one: A. 100 + 80 + 6 B. 1,000 + 80 + 6 C. 1,000,000 + 80,000 + 6,000 D. 1,000,000 + 800,000 + 60,000 Submit Answer